Physics 362K, Fall 2014
Test 3 Practice Questions
1. Two identical, spin-1/2 fermions of mass m are placed in a one-dimensional harmonic
oscillator potential, and interact with each other via a contact interaction. Their
Hamiltonian is
H=
p12 1
p2 1
+ m 2
Physics 362k, Fall 2014
Homework 6 Solutions
1. (15 points)
The Hamiltonian is
ge
1
1
ge
s
s ( B0 z + B1 (t ) x ) = B (s z B0 + s x B1 (t ) )
H = B = g e B s ( B0 z + B1 (t ) x ) = B
2
2
where x and z are the usual Pauli matrices. Writing them ou
Physics 362k, Fall 2014
Notes 3
I.
Hydrogen spectroscopy, electron spin and interactions
with fields, fine structure, and Zeeman effect
References 1, 2, 3, 4, 5, and 6 are sources for the historical material in this section of notes.
A.
Spectrum of the hy
Physics 362k, Fall 2014
First Day Review/Assessment Questions - Answers
One-dimensional wave mechanics
1. What is the value of Planck's constant h?
Answer: h 6.626 10 34 J s
2. What is the dispersion relation for matter waves?
Answer: In quantum mechanics
Physics 362K, Fall 2014
Quantum Physics II: Atoms and Molecules
2014 D. J. Heinzen
version 1
Contents
I.
Principles of Quantum Mechanics . 2
A.
Measurement and wave-particle duality . 2
1.
Blackbody radiation . 2
2.
Photoelectric effect . 5
3.
Momentum o
Physics 362k, Fall 2014
Homework 5 Solutions
1. (20 points) a) The wavefunction should have the following properties
i) It must have a zero at the origin and for all x < 0, since the potential is infinite for x 0.
ii) It must have no nodes, since it is t
Physics 362k, Fall 2014
Homework 4 Solutions
1. (30 points) Hydrogen atom fine structure
We have H fs = H kin + H so + H D ,
(1)
with
H kin
kinetic term:
p4
=
8me3c 2
(2)
1 1 V e 2 1 s
spin-orbit term:
H so
=
=
s
2 2
3
2me 2 c 2 r r
4e 0 2me c r
H
Physics 362k, Spring 2014
Homework 2 Solutions
1. (20 points) This problem can be done either with the analytical method, or with the algebraic
method starts with x
a a . I will give the analytical solution here.
2m
a) The first-order perturbation to th
Physics 362K, Fall 2014
Test 2 Practice Questions
1. Fine structure and Zeeman effect in hydrogen n = 4 states.
a) Write down all levels n 2 s +1 j belonging to the hydrogen n = 4 level, and give the
degeneracy of each level. (You do not have to list the
Grouped by j's: j = l + s
Efs equation on pg 7 on equation sheet
Bottom of pg 7 (weak Zee), nu = deltaE/hbar
pg 9 of equation sheet
Cla. Forbidden region
since V(x) > E
Hyperfine pg 8
omega_0 = deltaE/hbar
pg 10
Rabi Freq
Mag res, solve
for V from H
What
Physics 362K, Fall 2014
Test 1 Review Questions
This review contains more problems than will be on the test.
1. True or false
a. _ T _ F If O oi oi oi , then it follows that oi O oi oi .
b. _ T _ F
ai bi B A is the Hermitian conjugate of bi ai AB
c. _ T _
Physics 362K, Fall 2014
Test 2 Practice Solutions
1. Fine structure and Zeeman effect in hydrogen n = 4 states.
1
1
1
3
and .
2. a) With I = 1 and S = , the possible values of F are
2
2
2
b) From the formula sheet, the Fermi contact term is H Fermi
8 m
Physics 362K, Fall 2014
Test 1 Review Questions Solutions
This review contains more problems than will be on the test.
1. True or false
a. _ T _x_ F If O oi oi oi , then it follows that oi O oi oi .
If O is not Hermitian, then oi O is not the bra represen
Physics 362K, Fall 2014
Test 3 Practice Solutions
1.
1
2
2. a) If the two bosons are both in the state g their electronic state is elect = g 1 g 2 ,
and their total state is given by r = Rn (r )Ym ( , ) g 1 g 2 . Since the two particles
are identical boso
Physics 362k, Fall 2014
Homework 1 Solutions
1. (20 points) Relativistic particle dispersion relation.
If we substitute E h and p h / k into the relativistic energy formula, we obtain
E m2c 4 p 2c 2 mc 2
m2c 4 k c 2 mc 2
2
2
so the dispersion relation is
Physics 362k, Spring 2014
Homework 3 Solutions
1. (22 points) Hydrogen atom induced dipole moments and polarizability.
1
2
3
4
2. (16 points)
5
6
3. (30 points)
a) We can take the Hamiltonian as H H
(0)
H , where H
2
Vsq ( x, y ) ,
2m
2
(0)
with Vsq the